We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors, and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.

Figure 3.
Example 2: Top left: the time evolution of $\rho_1(x)$ solved from the original system (1) (blue solid lines) and the rescaled system (41) (red dashed lines). Top right: the time evolution of $u_1(x)$ solved from the original system (1) (blue solid lines) and the rescaled system (41) (red dashed lines). Bottom left: the distribution $f_1(x, v)$ at time $t = 0.7$ solved from the original system (1). Bottom right: the distribution $g_1(x, \xi)$ at time $t = 0.7$ solved from the rescaled system (41).

Figure 4.
Example 3: The density $\rho_\varepsilon (x)$ (left) and the macroscopic velocity $u_\varepsilon (x)$ (right) at time $t = 1$ computed by the scheme (42) with different $\varepsilon $'s are present, as well as that of the limiting system (6). The lines corresponding to $\varepsilon = 10^{-3}$ almost overlap with the lines of limiting system.

Figure 5.
Example 4: Time snapshots of the solution to the aggregation system. From left to right: the distribution $g_\varepsilon (x, \xi)$, the density $\rho_\varepsilon (x)$, the momentum $\rho_\varepsilon (x)u_\varepsilon (x)$ and the scaling factor $\omega_\varepsilon (x)$. In this test $\varepsilon = 1$.

Figure 6.
Example 4: Time snapshots of the solution to the aggregation system. From left to right: the distribution $g_\varepsilon (x, \xi)$, the density $\rho_\varepsilon (x)$, the momentum $\rho_\varepsilon (x)u_\varepsilon (x)$ and the scaling factor $\omega_\varepsilon (x)$. In this test $\varepsilon = 10^{-4}$. The stationary solution $\rho$ and $\rho u$ of the limiting system (6) is illustrated by red dashed lines in the last row.